A roped off area of width x is created around a 4-foot by 6-foot rectangular museum of display of Native American artifacts. The combined area of the display and the roped-off is 120 square feet. Find the width.
Answers
x----------> width a roped off area
we know that
(4+2x)*(6+2x)=120
then
4*6+4*2x+2x*6+2x*2x=120 ---> 24+8x+12x+4x²=120 -------> 4x²+20x+24=120
4x²+20x-96=0
using a graph tool---------> to calculate the quadratic equation
see the attached figure
the solution is
x=3 ft
the answer is
the width a roped off area is 3 ft
The width of the roped off area around the museum display is 4 feet.
To find the width of the roped off area around the museum display of Native American artifacts, we need to solve for x in the area equation. The area of just the display is 4 feet by 6 feet, which is 24 square feet. Since we know the combined area of the display and the roped-off area is 120 square feet, the equation is:
Area of display + Area of roped-off region = Total Area
(4)(6) + ((4 + 2x)(6 + 2x) - (4)(6)) = 120
24 + (24 + 4x + 12x + 4x2) - 24 = 120
Simplifying, we get:
4x² + 16x - 96 = 0
Divide by 4:
x² + 4x - 24 = 0
Factor the quadratic equation:
(x + 6)(x - 4) = 0
Setting each factor equal to zero gives us two potential solutions:
x + 6 = 0
or
x - 4 = 0
Therefore, x can be -6 or 4. Since a width can't be negative, the only logical solution is:
x = 4 feet
The width of the roped-off area is 4 feet.
Related Questions
Which function represents a line with a slope of −4 and a y-intercept of −2?
A) y = 4x − 2
B) y = −4x + 2
C) y = −4x − 2
D) y = −2x − 4
Answers
Answer:
C) [tex]y = -4x - 2[/tex]
Step-by-step explanation:
The general equation of a line in the xy plane is as follows:
[tex]y = mx + b[/tex]
where [tex]m[/tex] is the slope of the line, and [tex]b[/tex] is the intercept with the y axis.
In this case we want a line with a slope of -4, therefore:
[tex]m = -4[/tex]
and we also want a y-intercept of -2, so:
[tex]b = -2[/tex]
Substituting in the general equation of the straight line:
[tex]y = mx + b[/tex]
⇒[tex]y = -4x-2[/tex]
wich is option C
name the sine, cosine, and tangent of angle a in the figure below
Answers
now, we can simplify all those to
[tex]\bf cos(A)=\cfrac{24}{12\sqrt{5}}\implies \cfrac{2}{\sqrt{5}}\quad \stackrel{rationalizing~it}{\implies }\quad \cfrac{2\sqrt{5}}{5} \\\\\\ sin(A)=\cfrac{12}{12\sqrt{5}}\implies \cfrac{1}{\sqrt{5}}\quad \stackrel{rationalizing~it}{\implies }\quad\cfrac{\sqrt{5}}{5} \\\\\\ tan(A)=\cfrac{12}{24}\implies \cfrac{1}{2}[/tex]
Choose the correct simplification of the expression (a2)3. a5 a8 a a6
Answers
Answer:
The correct answer is A to the 6th power, A^6
Step-by-step explanation:
First, apply the exponent rule. (a^b)^c = a^bc
You would get a^2*3 (* means times) (^ means to the power)
Second refine.
You would get A^6.
The answer is A^6.
PLZ HELPP FAST
Solve the triangle. A=44 degrees, a = 7,b=8
Answers
Explanation:
1) Use the law of sines:
sin(A) sin(B)
----------- = --------- =>
a b
2) Solve for B:
sin(B) = b * sin(A) / a = 8 * sin(44°) / 7 = 0.79389528
=> B = arcsin(0.79389528) = 52.55°
3) A + B + C = 180° => C = 180° - A - B = 180° - 44° - 52.55° = 83.45°
4) Use the law of sines again:
sin(A) sin(C)
------------- = -------------
a c
5) Solve for c:
c = sin(C) * a / sin(A) = sin(83.45°) * 7 / sin(44°) = 10.0
Answer: a = 7, b = 8, c = 10.
If Martha earns $3.25 per hour for babysitting, how much will she earn in: 1.5 hours? 2.5 hours? 3.5 hours?
Answers
IN 1.5 HOURS=3.25*1.5=4.88 DOLLARS
IN 2.5 HOURS=3.25*2.5=8.13 DOLLARS
IN 3.5 HOURS=3.25*3.5=11.38 DOLLARS
Martha in:
1.5 hours will make: $4.88
2.5 hours will make: $8.13
3.5 hours will make: $11.38
Help with this question.
Answers
(x,y)
(cos[theta], sin[theta])
(4,-8)
So for cos to be POSITIVE, it can only be in either Quadrant I, or Quadrant IV
Quad I = (+,+)
Quad IV = (+,-) (Our cord has a negative sin, so that is it)
So it has to be in Quad IV, that is some information.
That narrows it down to either A or B.
If you have a calculator, you can put in:
cos(-2) or cos(-2/sq rt 5) i believe.... or its either the inverse, can’t entirely remember.
Im sorry I couldn’t help you more.
Maura is 5 years younger than her sister Cara. Seven years ago, Maura was half as old as her sister. How old is Maura now?
Answers
We know Maura is five years younger than Cara. In symbols that is, M - 5 = C
We also know that 7 years ago Maura's age was half of Cara's. Maura's age 7 years ago was M-7 and Cara's age seven years ago was C-7. Since Maura's age (7 years ago) was half of Cara's we can write in symbols: [tex]M-7= \frac{1}{2}(C-7) [/tex]
Let's take that last equation and substitute C - 5 for M (this is because according to our first equation these are equal). When we do this we get an equation with only C as the variable and solve for C as follows:
[tex]M-7= \frac{1}{2}(C-7) [/tex]
[tex]C-5-7= \frac{1}{2}(C-7) [/tex]
[tex]C-12= \frac{1}{2}C-( \frac{1}{2})(7) [/tex]
[tex]C-12= .5C-3.5[/tex]
[tex].5C-12= -3.5[/tex]
[tex].5C= 8.5[/tex]
[tex]C= 17[/tex]
Cara is 17. Since Maura is 5 years younger she is 12.
As a check, seven years ago Cara was 10 and Maura was 5. It is the case that Maura was half Cara's age seven years ago.
To solve this problem, set up equations based on the given information. M = C - 5 and M - 7 = (C - 7) / 2. Solve the system of equations to find Maura's age (19).
Explanation:To solve this problem, we can set up equations based on the given information. Let's let M represent Maura's age and C represent Cara's age.
The first clue tells us that Maura is 5 years younger than Cara, so we can write the equation M = C - 5.
The second clue tells us that seven years ago, Maura was half as old as Cara. So seven years ago, Maura's age was M - 7 and Cara's age was C - 7. We can write the equation M - 7 = (C - 7) / 2.
Now we can solve this system of equations to find Maura's age. Substituting the value of M from the first equation into the second equation, we get (C - 5) - 7 = (C - 7) / 2. Simplifying and solving for C, we find that Cara's current age is 24. Substituting this value back into the first equation, we find that Maura's current age is 19.
What needs to happen before you can use the reason CPCTC?
Answers
Explanation:
1) CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent.
2) Therefore, to use that reason you first need to prove that the the triangles are congruent.
3) Once, it has been stated that the triangles are congruent, then you can use the reason to state the the corresponding segments are congruent which will let you to state the length of any side of one triangle known the corresponding side of the other triangle, because they are congruent (have the same length).
For the Transformation T, write the T^-1.
T : (x, y) (5x, 5y )
T^ -1 (x, y)
(x - 5, y - 5)
(5x - 1, 5y - 1)
(1/5x, 1/5y)
Answers
___
(1/5)(5x) = x, so the inverse transformation returns the original. Not so for the other choices.
Answer: The correct option is
(C) [tex]T^{-1}(x,y)\rightarrow \left(\dfrac{1}{5}x,\dfrac{1}{5}y\right).[/tex]
Step-by-step explanation: We are given a transformation T defined as :
[tex]T:(x,y)\rightarrow (5x,5y).[/tex]
We are to find the inverse transformation [tex]T^{-1}(x,y).[/tex]
From the given transformation, we have
[tex]T:(x,y)\rightarrow (5x,5y)\\\\\Rightarrow T^{-1}(5x,5y)=(x,y).[/tex]
Let us consider that
[tex]5x=z~~~~~~\Rightarrow x=\dfrac{z}{5},\\\\\\5y=t~~~~~~\Rightarrow y=\dfrac{t}{5}.[/tex]
Therefore, we get
[tex]T^{-1}(5x,5y)\rightarrow(x,y)\\\\\Rightarrow T^{-1}(z,t)\rightarrow \left(\dfrac{z}{5},\dfrac{t}{5}\right)\\\\\\\Rightarrow T^{-1}(x,y)\rightarrow \left(\dfrac{1}{5}x,\dfrac{1}{5}y\right).[/tex]
Thus, (C) is the correct option.
How do I find the holes for this function?
Answers
x^2 - x - 2 factors into (x - 2)(x + 1) both of which can be crossed out or canceled with the factors in the numerator. That creates very really tiny holes at x = 2 and x = - 1.
[tex] {x}^{2} - x - 2 \: = > (x + 1)(x - 2)[/tex]
Which leaves is with the equation:
[tex]y = \frac{(x - 5)(x + 1)(x - 2)}{(x + 1)(x - 2)} [/tex]
We can see that we have the common factors of (x+1) and (x-2), so these cancel out in the numerator and denominator:
[tex]y = \frac{(x - 5)}{1} [/tex]
a.k.a: y = x - 5
So if x+1 and x-2 cancel out, then the x values of the holes are:
x+1=0
x= -1
And
x-2=0
x= 2
Now we plug in each of these numbers into our simplified equation:
y = x - 5
y=(-1)-5
y= -6
This gives us the coordinate (-1,-6)
y = x - 5
y=(2)-5
y= -3
This gives us the coordinate (2,-3)
So the holes are at (-1,-6) and (2,3)
The museum of science in boston estimated that 64% of all visitors came from within the state. on sunday, 2,500 people attended the museum. how many attended the museum from out of state?
Answers
Percentage of visitors who come from outside will be (100-64)=36%
The total number of attendees is 2500
the total number of of people who came from outside the state will be:
36/100×2500
=900 visitors
First, calculate the number of attendees from within the state by multiplying the total by 64% to get 1,600 people. Subtract this from the total to find the number of out-of-state attendees, 900 people.
Explanation:This is a problem of percentage utilization and subtraction. First, we must determine the number of in-state attendees by multiplying the total number of attendees (2500 people) by the given percentage of in-state visitors (64%).
So, 2500 people x 0.64 = 1600 people.
These are the people who attended from within the state. To find out how many visitors came from out of state, we subtract this number from the total: 2500 people - 1600 people = 900 people. Therefore, 900 people attended the museum from out of state on Sunday.
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A kite has a height of 36 inches and a width of 30 inches. Explain how to use the area formula for a triangle to find the area of the kite.
Answers
Height of the kite is = 36 inches.
Width of the kite is = 30 inches
One of the ways to find the area is to draw a vertical line to break the kite into two equal triangles. Mark the base as 36 inches and height as 15 inches .
Now we will use the formula [tex]area=\frac{1}{2}*base*height[/tex] to find the area of each triangle. Then we will add both the areas to find the area of the kite.
Area of 1 triangle = [tex]\frac{1}{2}*36*15[/tex] = 270 square inches
Area of the 2nd triangle is also = 270 square inches
Hence, area of the kite = 270+270 = 540 square inches
The area of the kite is 540 square inches
The dimension of the rectangle is given as:
Height = 36 inches. Width = 30 inches
The area of the kite is then calculated using:
[tex]Area = \frac 12 \times (Height \times Width)[/tex]
Substitute values for Height and Width
[tex]Area = \frac 12 \times (36 \times 30)[/tex]
Evaluate the expression in bracket
[tex]Area = \frac 12 \times 1080[/tex]
Evaluate the product
[tex]Area = 540[/tex]
Hence, the area of the kite is 540 square inches
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The elevator in the Empire State Building can pass 80 floors in 45 seconds. How many floors can it pass in 9 seconds?
Answers
Which properties of equality justify steps b and d?
A. Multiplication Property of Equality; Subtraction Property of Equality
B. Subtraction Property of Equality; Division Property of Equality
C. Subtraction Property of Equality; Multiplication Property of Equality
D. Subtraction Property of Equality; Subtraction Property of Equality
Answers
For the first one, you are subtracting 7 from both sides (step b), to help isolate the y
since you are both subtracting, and "both sides" (meaning there is a equal sign) it is Subtraction Property of Equality.
For the second one, you are multiplying 2 from both sides (step d), to help isolate the y
since you are multiplying, and "both sides" (meaning there is a equal sign), it is Multiplication Property of Equality
hope this helps
C) Subtraction Property of Equality; Multiplication Property of Equality
- Prodixy ☕
PLEASE someone help me on this
I would appreciate it.
Answers
multiply by 4
8x - 4y = 12
3x +4y = 10 (unchanged equation)
------------------add
11x = 22
answer is A. the first.
first equation multiplied by 4 then sum to get 11x = 22
6s - 4 = 8 (2 + 1/4s)
Answers
6s-4=16+2s
-2s -2s
4s-4=16
+4 +4
4s=20
s=5
To solve the equation 6s - 4 = 8 (2 + 1/4s), simplify and isolate the variable 's'. The solution is s = 5.
Explanation:To solve the equation 6s - 4 = 8 (2 + 1/4s), we need to simplify and isolate the variable 's'. First, distribute the 8 to both terms inside the parentheses:
6s - 4 = 16 + 2s
Next, move the variable terms to one side of the equation and the constant terms to the other side by subtracting 2s from both sides and adding 4 to both sides:
6s - 2s = 16 + 4
Combine like terms to simplify:
4s = 20
Finally, divide both sides by 4 to solve for 's':
s = 5
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A triangular lawn has sides that are 8 m and 12 m long with an included angle of 51°.
What is the area of this lawn?
Enter your answer as a decimal in the box. Round only your final answer to the nearest tenth.
Answers
.. A = (1/2)ab*sin(C)
.. = (1/2)(8 m)(12 m)sin(51°)
area ≈ 37.3 m^2
what is the area of this triangle 13in 94 degrees 7.5 in
Can you check my answer?
Which of the following is an example of why irrational numbers are 'not' closed under addition?
√4 + √4 = 2 + 2 = 4, and 4 is not irrantonal
1/2 + 1/2 = 1, and 1 is not irrational
√10 + (-√10) = 0, and 0 is not irrational
-3 + 3 = 0, and 0 is not irrational
I was thinking:
–3 + 3 = 0, and 0 is not irrational
because it came up with a different number besides 3.,
Answers
-3 = -3/1
3 = 3/1
0 = 0/1
each can be written as a fraction of whole numbers, so that's why they are rational
The actual answer is choice C
We are adding the square root of 10, written sqrt(10) in shorthand, to the negative version of the same number. Doing so leads to 0. This is using the property x + (-x) = 0. The left hand side of choice C has two irrational numbers. They add to 0 on the right hand side which is rational. The fact that we added two irrational numbers to get an rational result indicates that irrational numbers are not closed under addition.
As viewed from a cliff 340 ft above sea level a ship is 450 feet from the shore what is the angle of depression from the cliff to the ship on the water below
Answers
To get the angle of depression we use the formula:
tan x=opposite/adjacent
where x is the angle of elavation
opposite is the height of the cliff
adjacent is the distance of the ship from the shore
thus
tan x=340/450
tan x=34/45
x=arctan 34/45
x=37.07
hence the angle of elevation= angle of depression=37.1°
Solve the equation. Show all steps and work. Then check your answers (show all work). Finally, write the solution set.
x+1=√9x-11
Answers
[tex]x+1=\sqrt{9x-11}\\ (x+1)^2=9x-11\\\ x²+2x+1-9x+11=0\\ x^2-7x+12=0\\ x²-4x-3x+12=0\\ x(x-4)-3(x-4)=0\\ (x-4)(x-3)=0\\ x=4\ or\ x=3 [/tex]
Sol={3,4}
(4,3) is the required solution set for the given equation
Given equation is x+1=√9x-11
Thus solving above equation for x :
[tex]x+1=\sqrt{9x-11}\\[/tex]
Squaring throughout we get,
[tex](x+1)^{2} =9x-11\\x^{2} +2x+1=9x-11\\x^2+2x-9x+1+11=0\\x^2-7x+12=0\\x^2-3x-4x+12=0\\x(x-3)-4(x-3)=0\\(x-4)(x-3)=0\\[/tex]
Thus we get :
Either (x-4) = 0 or (x-3) = 0
Thus we have x = 4 or x = 3
Hence x = 4 or x = 3 is the solution set
Phil is a commercial fisherman. He keeps 5 salmon for every 8 cod he catches. If Phil keeps 15 salmon, how many cod will he keep?
Answers
3 * 8 = 24 cod
The weights, in pounds, of a group of student are as follows: 173 123 171 175 188 120 177 160 151 169 162 128 145 140 158 132 202 162 154 180 164 166 157 171 175 Determine the mean, standard deviation, and five-number summary for the data. a. Mean is 160.12; Standard deviation is 19.8. Five-number summary: min = 120, 4072-02-01-06-00_files/i0290000.jpg= 148, median = 162, 4072-02-01-06-00_files/i0290001.jpg= 174, max = 202 b. Mean is 162; Standard deviation is 19.8. Five-number summary: min = 120, 4072-02-01-06-00_files/i0290002.jpg= 148, median = 160.12, 4072-02-01-06-00_files/i0290003.jpg= 174, max = 202 c. Mean is 160.12; Standard deviation is 15.5. Five-number summary: min = 202, 4072-02-01-06-00_files/i0290004.jpg= 148, median = 160.12, 4072-02-01-06-00_files/i0290005.jpg= 174, max = 120 d. Mean is 162; Standard deviation is 15.5. Five-number summary: min = 202, 4072-02-01-06-00_files/i0290006.jpg= 148, median = 162, 4072-02-01-06-00_files/i0290007.jpg= 174, max = 120
Answers
Standard Deviation: 19.76
Median: 162
Maximum: 202
Minimum: 120
The mean is calculated by adding all of the numbers in the data set and dividing by the number of values that were added:
173 + 123 + 171 + 175 + 188 + 120 + 177 + 160 + 151 + 169 + 162 + 128 + 145 + 140 + 158 + 132 + 202 + 162 + 154 + 180 + 164 + 166 + 157 + 171 + 175 ÷ 25 = 160.12
Standard deviation is found by calculating the mean (160.12) , and then subtracting the mean and square root result for each number. Finally, working out the mean of the squared differences and taking the square root of the final answer results in a standard deviation of 19.76. (The attached image has the formula for finding standard deviation).
Median is the number that is found in the middle when placing the numbers in order from least to greatest.
The maximum is the largest number in the data set.
The minimum is the smallest number in the data set.
99 POINTS!!!!! FAST
Which set represents the domain of the function shown?
{(−3, 6), (0, 2), (4, 7), (11, 15)}
A. {(6, −3), (2, 0), (7, 4), (15, 11)}
B. {2, 6, 7, 15}
C. {−3, 0, 4, 11}
D. {−3, 0, 2, 4, 6, 7, 11, 15}
(P.S. The 99 points were obviously fake XD)
Answers
The Domain of the function {(−3, 6), (0, 2), (4, 7), (11, 15)} is
What is Domain of Function?The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f(x) make up this set. A function's range is the collection of values it can take as input.
Given:
Set :{(−3, 6), (0, 2), (4, 7), (11, 15)}
So, the Domain is the input value
Domain = { -3, 0, 4, 11}
and, the range is output value
Range= {6, 2, 7, 15}
Hence, the domain is { -3, 0, 4, 11}.
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find the area of the circle .Leave your answer in terms of pie (3.14) the area is 4.1m .I need help on this question
Answers
Answer:
Area of the circle = 1.3π m²
Step-by-step explanation:
Area of the circle is given by the formula A = πr²
It is given that area of the circle is 4.1 m² and we have to convert it in the terms of π.
Now to convert the area in terms of π we will multiply the area by π and divide the area by 3.14.
⇒ Area×π / 3.14 = 4.1 × π/3.14 = 1.30π
Therefore answer is area of the circle = 1.30π m²
5 times the sum of p and r
A.5(p+r)
B.5p+r
C.5pr
D.5+p+r
Answers
Option: A is the correct answer.
The mathematical representation of the given statement is:
[tex]5(p+r)[/tex]
Step-by-step explanation:We are asked to represent the given expression in words to numerical expression.
The statement is:
5 times the sum of p and r
The sum of p and r is: (p+r)
and now 5 times the sum of p and r i.e. 5 is multiplied to the sum of p and r.
i.e.
[tex]5(p+r)[/tex]
All four functions below are quadratic functions. Which function has the greatest maximum value? Answer options attached in pictures. (10 points!!)
Answers
Factor completely, then place the answer in the proper location on the grid. a ^2 - 25b ^2
Answers
Answer:
[tex](a+5b)(a-5b)[/tex]
Step-by-step explanation:
[tex]a^2 - 25b ^2[/tex]
Factor completely
write 25 in square form , 25= 5^2
[tex]a^2 -5^2b ^2[/tex]
[tex]a^2 -(5b)^2[/tex]
Apply difference of squares formula to factor it
[tex]x^2 - y^2 = (x+y)(x-y)[/tex]
[tex]a^2 -(5b)^2[/tex]
x= a and y=5b
[tex](a+5b)(a-5b)[/tex]
Which of the following ordered pairs is a solution to y = 2x?
(3, 6)
(3, 8)
(3, 9)
(3,27)
Answers
Answer:
The answer is (3, 8)
Step-by-step explanation:
so next time you try to make a exponent use ^ exsample 3^9 3 to the power of 9 anyways since the x axis or sum is 3 it is y = 2^3 multiply it so 2 x 2 x 2 = 8 so that answer is (3, 8)
if correct please crown
A manufacture of skateboards offered a 5/2/1 chain discount too many customers. Bobs sporting goods ordered 20 skateboards for a total of $625 list price. What was the net price of the skateboards?
Answers
1) Apply 5% discount to the original price.
2) Apply 2% discount to the result of step 1).
3) Apply 1% discount to the result of step 2).
We have then that the result will be given by:
625 * (0.95) * (0.98) * (0.99) = 576.05625 $
Then, the price of each skateboard is:
(576.05625) /20=28.8028125 $
Answer:
The net price of the skateboards was:
28.8028125 $
A population of 80,000 toads is expected to shrink at a rate of 9.2% per year.
What will the toad population be in 20 years?
3632
7360
11,609
72,640,